We assume that human voluntary rhythmic motion in variable gravity \(g(t)\)  may be seen as a dynamical system with bounded motion in phase space, described by a separable Hamiltonian \(H(I_\alpha , \theta ^\alpha ,\lambda (t))=\sum ^D_{\alpha =1}H_j(I_\alpha , \theta ^\alpha ,\lambda (t))\) . The Hamiltonian depends on action-angle variables \(I_\alpha \)  and \(\theta ^\alpha \)  respectively and on a time-dependent function \(\lambda (t)\) , accounting for the modifications induced by \(g(t)\) . The various ingredients underlying the latter assumption deserve further comments. First, Hamiltonian dynamics being the most powerful formulation of classical Mechanics, it is rather natural to adopt a Hamiltonian approach.
This being said, not every dynamics is Hamiltonian: A sufficient criterion for a Hamiltonian to exist is that the total time-derivative applied to the Poisson bracket of two functions defined on phase-space obeys the Leibniz rule [1]}.
The latter criterion cannot a priori be checked from our experimental setup: We will a posteriori confirm
that the observed phase-space trajectories are not incompatible with a Hamiltonian dynamics. Second, the dynamics has a priori no reason to be separable. The main interest we have in
assuming the separability is that it allows a clear separation between vertical and horizontal
directions (with respect to \(\vec{g}\) ), the dynamics in the vertical direction being intuitively the
most strongly impacted by the variations of \(g\,\) . Again, the separability hypothesis will only be checked a posteriori by observing the phase-space trajectories, see next Section.
